I have the following recursion relation:
$d(m,k)=d(m-1,k) + d(m-1,k-1) + ... + d(m-1,k-\min(k+1,m)+1); \hspace{2cm} m=1,2,3,...; k=0,1,2,...,\binom{m}{2}.$
with the following conditions
$d(m,0) = 1, d(1,k) = 0 \text{ for } k>0.$
For $k \leq m-1,$ the recursion becomes
$d(m,k)=d(m,k-1)+d(m-1,k).$
How to get a closed form of $d(m,k)$ for any $k$?